Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Constant Dimension Codes Anna-Lena Trautmann Institute of Mathematics University of Zurich “Crypto and Coding” Z¨ urich, March 12th 2012 1 / 25
Isometry and Automorphisms of Constant Dimension Codes Introduction 1 Introduction 2 Isometry of Random Network Codes 3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes 2 / 25
Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding 3 / 25
Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes 3 / 25
Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding 3 / 25
Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation constant dimension codes are used for random network coding isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes 3 / 25
Isometry and Automorphisms of Constant Dimension Codes Introduction Random Network Codes Definition The projective geometry P ( F n q ) is the set of all subspaces of F n q . A random network code is a subset of P ( F n q ). 4 / 25
Isometry and Automorphisms of Constant Dimension Codes Introduction Random Network Codes Definition The projective geometry P ( F n q ) is the set of all subspaces of F n q . A random network code is a subset of P ( F n q ). Definition Subspace metric: d S ( U , V ) = dim( U + V ) − dim( U ∩ V ) Injection metric: d I ( U , V ) = max(dim U , dim V ) − dim( U ∩ V ) 4 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes 1 Introduction 2 Isometry of Random Network Codes 3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes 5 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Definition A distance-preserving map ι : P ( F n q ) → P ( F n q ) i.e. fulfilling ∀ U , V ∈ P ( F n d ( U , V ) = d ( ι ( U ) , ι ( V )) q ) . is called an isometry on P ( F n q ). 6 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Definition A distance-preserving map ι : P ( F n q ) → P ( F n q ) i.e. fulfilling ∀ U , V ∈ P ( F n d ( U , V ) = d ( ι ( U ) , ι ( V )) q ) . is called an isometry on P ( F n q ). Any isometry ι is injective: U � = V ⇐ ⇒ d ( U , V ) � = 0 ⇐ ⇒ d ( ι ( U ) , ι ( V )) � = 0 ⇐ ⇒ ι ( U ) � = ι ( V ) and hence, if the domain is equal to the codomain, bijective. The inverse map ι − 1 is an isometry as well. 6 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Lemma If ι : P ( F n q ) → P ( F n { 0 } , F n � � q ) is an isometry, then ι ( { 0 } ) ∈ . q 7 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Lemma If ι : P ( F n q ) → P ( F n { 0 } , F n � � q ) is an isometry, then ι ( { 0 } ) ∈ . q Lemma Let ι be as before and U ∈ P ( F n q ) arbitrary. Then ι ( { 0 } ) = { 0 } = ⇒ dim( U ) = d ( { 0 } , U ) = d ( { 0 } , ι ( U )) = dim( ι ( U )) and on the other hand ι ( { 0 } ) = F n ⇒ dim( U ) = d ( { 0 } , U ) = d ( F n q = q , ι ( U )) = n − dim( ι ( U )) . 7 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Lemma If ι : P ( F n q ) → P ( F n { 0 } , F n � � q ) is an isometry, then ι ( { 0 } ) ∈ . q Lemma Let ι be as before and U ∈ P ( F n q ) arbitrary. Then ι ( { 0 } ) = { 0 } = ⇒ dim( U ) = d ( { 0 } , U ) = d ( { 0 } , ι ( U )) = dim( ι ( U )) and on the other hand ι ( { 0 } ) = F n ⇒ dim( U ) = d ( { 0 } , U ) = d ( F n q = q , ι ( U )) = n − dim( ι ( U )) . The isometries with ι ( { 0 } ) = { 0 } are exactly the isometries that keep the dimension of a codeword. 7 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Theorem (Fundamental Theorem of Projective Geometry) Every order-preserving bijection f : P ( F n q ) → P ( F n q ) , where n > 2 , is induced by a semilinear transformation ( A, α ) ∈ PΓL n = (GL n / Z n ) ⋊ Aut( F q ) where Z n = { µI n | µ ∈ F ∗ q } is the set of scalar transformations. 8 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Theorem (Fundamental Theorem of Projective Geometry) Every order-preserving bijection f : P ( F n q ) → P ( F n q ) , where n > 2 , is induced by a semilinear transformation ( A, α ) ∈ PΓL n = (GL n / Z n ) ⋊ Aut( F q ) where Z n = { µI n | µ ∈ F ∗ q } is the set of scalar transformations. Theorem For n > 2 a map ι : P ( F n q ) → P ( F n q ) is an order-preserving bijection (with respect to the subset relation) of P ( F n q ) if and only if it is an isometry with ι ( { 0 } ) = { 0 } . 8 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Definition 1 Two codes C 1 , C 2 ⊆ P ( F n q ) are linearly isometric if there exists A ∈ PGL n (or GL n ) such that C 1 = C 2 A . 2 We call C 1 and C 2 semilinearly isometric if there exists ( A, α ) ∈ PΓL n (or ΓL n ) such that C 1 = C 2 ( A, α ). 9 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Definition 1 Two codes C 1 , C 2 ⊆ P ( F n q ) are linearly isometric if there exists A ∈ PGL n (or GL n ) such that C 1 = C 2 A . 2 We call C 1 and C 2 semilinearly isometric if there exists ( A, α ) ∈ PΓL n (or ΓL n ) such that C 1 = C 2 ( A, α ). Remark: 1 All isometric codes are equivalent from a coding point of view (i.e. same rate and error correction capability). 9 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes Definition 1 Two codes C 1 , C 2 ⊆ P ( F n q ) are linearly isometric if there exists A ∈ PGL n (or GL n ) such that C 1 = C 2 A . 2 We call C 1 and C 2 semilinearly isometric if there exists ( A, α ) ∈ PΓL n (or ΓL n ) such that C 1 = C 2 ( A, α ). Remark: 1 All isometric codes are equivalent from a coding point of view (i.e. same rate and error correction capability). 2 There are more equivalence maps than order-preserving isometries. 9 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions 1 Introduction 2 Isometry of Random Network Codes 3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes 10 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Definition For a given code C ⊆ P ( F n q ), Aut( C ) = { A ∈ GL n |C A = C} is called the (linear) automorphism group of the code. 11 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Definition For a given code C ⊆ P ( F n q ), Aut( C ) = { A ∈ GL n |C A = C} is called the (linear) automorphism group of the code. Definition The Grassmannian G q ( k, n ) is the set of all k -dimensional subspaces of F n q . A constant dimension code is a subset of G q ( k, n ). 11 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes 1 Introduction 2 Isometry of Random Network Codes 3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes 12 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes Theorem All Desarguesian spread codes are linearly isometric. 13 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes Theorem All Desarguesian spread codes are linearly isometric. Proof: Since there is only one spread of lines in F l q k , different Desarguesian spreads of F n q can only arise from the different isomorphisms between F q k and F k q . As the isomorphisms are linear maps, there exists a linear map between the different spreads arising from them. 13 / 25
Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes Theorem The linear automorphism group of a Desarguesian spread code k ( q k ) × Aut( F q k ) . C ⊆ G q ( k, n ) is isomorphic to GL n 14 / 25
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